applied fracture mechanics

Chapter 3Fractal Fracture MechanicsApplied to Materials EngineeringLucas Máximo Alves and Luiz Alkimin de LacerdaAdditional information is available at the end of the chapterhttp://dx.doi.org/10.5772/525111. IntroductionThe Classical Fracture Mechanics (CFM) quantifies velocity and energy dissipation of acrack growth in terms of the projected lengths and areas along the growth direction.However, in the fracture phenomenon, as in nature, geometrical forms are normallyirregular and not easily characterized with regular forms of Euclidean geometry. As anexample of this limitation, there is the problem of stable crack growth, characterized by theJ-R curve [1, 2]. The rising of this curve has been analyzed by qualitative arguments [1, 2, 3,4] but no definite explanation in the realm of EPFM has been provided.Alternatively, fractal geometry is a powerful mathematical tool to describe irregular andcomplex geometric structures, such as fracture surfaces [5, 6]. It is well known fromexperimental observations that cracks and fracture surfaces are statistical fractal objects [7, 8,9]. In this sense, knowing how to calculate their true lengths and areas allows a morerealistic mathematical description of the fracture phenomenon [10]. Also, the differentgeometric details contained in the fracture surface tell the history of the crack growth andthe difficulties encountered during the fracture process [11]. For this reason, it is reasonableto consider in an explicit manner the fractal properties of fracture surfaces, and manyscientists have worked on the characterization of the topography of the fracture surfaceusing the fractal dimension [12, 13]. At certain point, it became necessary to include thetopology of the fracture surface into the equations of the Classical Fracture Mechanicstheory [6, 8, 14]. This new “Fractal Fracture Mechanics” (FFM) follows the fundamentalbasis of the Classical Fracture Mechanics, with subtle modifications of its equations andconsidering the fractal aspects of the fracture surface with analytical expressions [15, 16].The objective of this chapter is to include the fractal theory into the elastic and plastic energyreleased rates G0and J0, in a different way compared to other authors [8, 13, 14, 17, 18, 19].The non-differentiability of the fractal functions is avoided by developing a differentiable© 2012 Alves and de Lacerda, licensee InTech. This is an open access chapter distributed under the terms ofthe Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.